Remark 4.15 
This remark is essentially from [1]}. Let \(k\)  be a commutative ring with 1, and let \(\mathfrak {G}\)  be a reductive \(k\) -group. Hereupon, we denote by \(\kappa (s)\)  the residue field of \(s\)  and \(\bar{\kappa }(s)\)  the algebraic closure of \(\kappa (s)\) , for all \(s\in \operatorname{Spec}k\) .
A subgroup \(\mathfrak {P}\)  of \(\mathfrak {G}\)  is a parabolic (resp. Borel) subgroup if \(\mathfrak {P}\)  is smooth and \(\mathfrak {P}_{\bar{\kappa }(s)}\)  is a parabolic (resp. Borel) subgroup of \(\mathfrak {G}_{\bar{\kappa }(s)}\) , for all \(s\in \operatorname{Spec}k\) .
